Without doing a serious CFD project does anyone know of a back of the envelope calculation or model that will let me show the following. Given:
- Fluid Velocity (uniform, incompressible)
- Fluid Density
- Kinematic Viscosity Diameter of a sphere (porous)
- Porosity of sphere (blockage ratio)
I would like to know the fluid velocity inside the sphere at various “latitudes” in the sphere. I would only need to know 3 to 4 graduations in say the top hemisphere (I am assuming symmetry). If the flow is coming from left to right the part of the sphere normal will have a higher porosity because the holes will be normal to the current or a 90 deg angle of attack. As one gets closer to the edge the holes will appear smaller and have less porosity because their angle of attack is much less.
Re: Fluid Flow through a Porous Sphere
Unsure about the description of your problem. Are the pores small compared to the sphere itself? Are you thinking of Darcy's flow within the sphere?
In reply to Re: Fluid Flow through a Porous Sphere by Zhigang Suo
Re: More on fluid flow thru a porous sphere
Just to add a bit to Zhigang's comment; in the original post a value of porosity is assumed to be given. Porosity does not necessarily imply permeability. I assume that permeability is meant. Also assuming that the sphere is large enough that homogenity can be assumed and that the flow has a low Re number, one can use Darcy's law
u_i = K_{ij}/mu dp/dx_j
where u is the velocity, K is the permeability tensor, mu is the viscosity of the fluid, p is the pressure, x is the spatial direction.
A rough and ready 1-D version is
Q/A = K/mu delta p/L
where Q is the flow rate through an area A and delta p is the pressure drop over a length L.
For high Re number flows use the Forchheimer equation
Q/A + alpha (Q/A)^m = K/mu delta p/L
or the Ergun equation.
The pressure drop can be approximated assuming that the sphere is impermeable (look for the fluid dynamics of a raindrop for an example).
On the other hand, if the holes are large then the problem can be treated as a laminar flow through pipe with the input velocity distribution determined by the location of the hole relative to the circumference of the sphere.
-- Biswajit